The document discusses taxicab geometry, a form of non-Euclidean geometry focusing on the concept of distance defined as the sum of the absolute differences of coordinates. It highlights applications in urban settings, such as determining school district boundaries, and emphasizes its pedagogical value in teaching geometry. Key figures in the development of taxicab geometry, including Hermann Minkowski and Karl Menger, are mentioned along with references to geometric figures like circles and parabolas.

1. Taxicab Geometry A Study Into Non-Euclidean GeometryUsing Geometer SketchpadTyler RoellStudied under Dr. R. TalbertFranklin College Math DayNovember 21, 2009

2. The Beginning of Taxicab Geometry?Hermann Minkowski (1864-1909)

3. Teacher of Einstein

4. Also known as Manhattan Distance

5. Karl Menger (1952)What is Taxicab Geometry?Assumptions madeof Taxicab GeometryOne Main Difference

6. Dt=|x1-x2|+|y1-y2|DifferencesDt=|x1-x2|+|y1-y2|

7. De=

8. Crows Fly Vs. Urban GeographyDt=|3-1|+|1-4|=5

9. Redistricting ProblemSay you have three schools, K,N, and M. Using the Taxicab Distance, where should the school districts be drawn to keep all the districts the same size in area?

10. Geometric FiguresCircles

11. The set of all points equidistant from a point p with radius rGeometric FiguresParabolas

12. set of all points that are the same distance from a fixed line and a fixed point not on the lineGeometric FiguresParabolas

13. set of all points that are the same distance from a fixed line and a fixed point not on the lineBisectorsIn EuclideanIn Taxicab

14. Now back to the problemWhat should be done first?

15. What figures should be used?

16. Construct circles with the same radius

17. Construct the bisector between K and N

18. Bisector between N and M

19. Bisector between K and M

20. The Solution

21. Further ApplicationEllipses

22. The set of all points where the sum of the distances from any point on the curve to the two fixed points remain a constant Applications for TeachingEasier way to measure distanceApplicable to the Urban SettingUses technology in the Classroom

23. Book UsedEugene F. Krause Taxicab Geometry: An Adventure in Non-Euclidean Geometry